average utility is given byU2.3Because it falls short ofU3, we can assume that the person will refuse to make the $5,000 bet. Finally, the utility of the $15,000 bet is the average of the utility from $50,000 and the utility from $20,000. This is given byU1, which falls belowU2. In other words, the person likes the risky $15,000 bet even less than the $5,000 bet.
F I G U R E 4 . 1
R i s k A v e r s i o n
U U3
U2 U1
Income (thousands of dollars)
0 20 30 3335 40 50
Utility
An individual characterized by the utility-of-income curveUwill obtain a higher utility (U3) from a risk-free income of $35,000 than from a 50-50 chance of winning or losing $5,000 (U2). He or she will be willing to pay up to $2,000 to avoid having to take this bet. A fair bet of $15,000 provides even less utility (U1) than the $5,000 bet.
3This average utility can be found by drawing the chord joiningU($40,000) andU($30,000) and finding the midpoint of that chord. Because the vertical line at $35,000 is midway between $40,000 and $30,000, it will also bisect the chord.
KEEPinMIND
Choosing among Gambles
To solve problems involving a consumer’s choice over gambles, you should proceed in two steps. First, using the formula for expected values, compute the consumer’s expected utility from each gamble.
Then choose the gamble with the highest value of this number.
Willingness to Pay to Avoid Risk
Diminished marginal utility of income, as shown in Figure 4.1, means that people will be averse to risk. Among outcomes with the same expected dollar values ($35,000 in all of our examples), people will prefer risk-free to risky ones because the gains such risky outcomes offer are worth less in utility terms than the losses. In fact, a person would be willing to give up some amount of income to avoid taking a risk. In Figure 4.1, for example, a risk-free income of $33,000 provides the same utility as the $5,000 gamble (U2). The individual is willing to pay up to $2,000 to avoid taking that risk. There are a number of ways this person might spend these funds to reduce the risk or avoid it completely, which we will study below. Saying that someone is ‘‘very risk averse’’ is the same as saying that he or she is willing to spend a lot to avoid risk.
The shape of the utility-of-income curve, such asUin Figure 4.1, provides some idea of how risk averse the individual is. IfUbends sharply, then the utility the individual obtains from a certain out- come will be well above the expected utility from an uncertain gamble with the same expected payoff.
The lessUbends (that is, the more linearUis), the less risk averse is the person. In the extreme, ifUis a straight line, then the person will be indifferent between a certain outcome and a gamble with the same expected payoff. In other words, he or she would accept any fair gamble. A person with these risk preferences is said to berisk neutral.
Even for a very risk-averse person with a utility-of-income curve that is sharply bent as in Figure 4.1, if we took a small piece of the curve, say that between incomes
$33,000 and $35,000, and blew it up to be able to see it better, this piece looks almost like a straight line. Because straight lines are associated with risk-neutral individuals, this graphical exercise suggests that even people who are risk averse over large gambles (with, say, thousands of dollars at stake) will be nearly risk neutral over small gambles (with only a few dollars at stake). People are not very averse to small risks because even the worst case with a small risk does not reduce the person’s income appreciably.
METHODS FOR REDUCING RISK AND UNCERTAINTY
In many situations, taking risks is unavoidable. Even though driving your car or eating a meal at a restaurant subjects you to some uncertainty about what will actually happen, short of becoming a hermit, there is no way you can avoid every risk in your life. Our analysis in the previous section suggests, however, that people are generally willing to pay something to reduce these risks. In this section, we examine four methods for doing so—insurance, diversification, flexibility, and information acquisition.
M i c r o Q u i z 4 . 2
What would the utility-of-income curveUbe shaped like for someone who prefers risky situations?
Risk neutral
Willing to accept any fair gamble.
Insurance
Each year, people in the United States spend more than half a trillion dollars on insurance of all types. Most commonly, they buy coverage for their own life, for their home and automobiles, and for their health care costs. But, insurance can be bought (perhaps at a very high price) for practically any risk imaginable. For example, many people in California buy earthquake insurance, outdoor swimming pool owners can buy special coverage for injuries to falling parachutists, and surgeons or basketball players can insure their hands. In all of these cases, people are willing to pay a premium to an insurance company in order to be assured of compensation if something goes wrong.
The underlying motive for insurance purchases is illustrated in Figure 4.2.
Here, we have repeated the utility-of-income curve from Figure 4.1, but now we assume that during the next year this person with a $35,000 current income (and consumption) faces a 50 percent chance of having $15,000 in unexpected medical bills, which would reduce his or her consumption to $20,000. Without insurance, this person’s utility would beU1—the average of the utility from $35,000 and the utility from $20,000.
F I G U R E 4 . 2
I ns u r a nc e Re du c es R i s k
U U2
U0 U1
Income (thousands of dollars)
0 20 22 24 27.5 35
Utility
A person with $35,000 in income who faced a 50-50 chance of $15,000 in medical bills would have an expected utility ofU1. With fair insurance (which costs $7,500), utility would beU2. Even unfair insurance costing $11,000 would still yield the same utility (U1) as facing the world uninsured. But a premium of $13,000, which provides a utility of onlyU0, would be too costly.
Fair Insurance This person would clearly be better off with an actuariallyfair insurancepolicy for his or her health care needs. This policy would cost $7,500—
the expected value of what insurance companies would have to pay each year in health claims. A person who bought the policy would be assured of $27,500 in consumption. If he or she bought the policy and stayed well, income would be reduced by the $7,500 premium. If this person suffered the illness, the insurance company would pay the $15,000 in medical bills but this person would have paid the $7,500 premium so consumption would still be $27,500. As Figure 4.2 shows, the utility from a certain income of $27,500 (U2) exceeds that attainable from facing the world uninsured, so the policy represents a utility-enhancing use for funds.
Unfair Insurance No insurance company can afford to sell insurance at actua- rially fair premiums. Not only do insurance companies have to pay benefits, but they must also maintain records, collect premiums, investigate claims to ensure they are not fraudulent, and perhaps return a profit to shareholders. Hence, a would-be insurance purchaser can always expect to pay more than an actuarially fair premium. Still, a buyer may decide that the risk reduction that insurance provides is worth the extra charges. In the health care illustration in Figure 4.2, for example, this person would be willing to pay up to $11,000 for health insurance because the risk-free consumption stream of $24,000 that buying such ‘‘unfair’’ insurance would yield provides as much utility (U1) as does facing the world uninsured. Of course, even a desirable product such as insurance can become too expensive. At a price of $13,000, the utility provided with full insurance (U0) falls short of what would be obtained from facing the world uninsured. In this case, this person is better off taking the risk of paying his or her own medical bills than accepting such an unfair insurance premium. In Application 4.2: Deductibles in Insurance, we look at one way to avoid unfair insurance associated with small risks.
Uninsurable Risks The preceding discussion shows that risk-averse individuals will always buy insurance against risky outcomes unless insurance premiums exceed the expected value of a loss by too much. Three types of factors may result in such high premiums and thereby cause some risks to become uninsurable. First, some risks may be so unique or difficult to evaluate that an insurer may have no idea how to set the premium level. Determining an actuarially fair premium requires that a given risky situation must occur frequently enough so that the insurer can both estimate the expected value of the loss and rely on being able to cover expected payouts with premiums from individuals who do not suffer losses.
For rare or very unpredictable events such as wars, nuclear power plant mishaps, or invasions from Mars, would-be insurers may have no basis for establishing insur- ance premiums and therefore refrain from offering any coverage.
Two other reasons for absence of insurance coverage relate to the behavior of the individuals who want to buy insurance. In some cases, these individuals may know more about the likelihood that they will suffer a loss than does an insurer.
Those who expect large losses will buy insurance, whereas those who expect small ones will not. Thisadverse selectionresults in the insurer paying out more in losses than expected unless the insurer finds a way to control who buys the policies offered. As we will see later, in the absence of such controls, no insurance would be provided even though people would willingly buy it.
Fair insurance
Insurance for which the premium is equal to the expected value of the loss.
A P P L I C A T I O N 4 . 2 Deductibles in Insurance
A ‘‘deductible’’ provision in an insurance policy is the require- ment that the insured pay the firstXdollars in the event of a claim; after that, insurance kicks in. With automobile insurance policies, for example, a $500 deductible provision is quite standard. If you have a collision, you must pay the first $500 in damages, then the insurance company will pay the rest.
Most other casualty insurance policies have similar provisions.
Deductibles and Administrative Costs
The primary reason for deductible provisions in insurance contracts is to deter small claims. Because administrative costs to the insurance company of handling a claim are about the same regardless of a claim’s size, such costs will tend to be a very high fraction of the value of a small claim.
Hence, insurance against small losses will tend to be actua- rily ‘‘unfair.’’ Most people will find that they would rather incur the risks of such losses (such as scratches to the finish of their cars) themselves rather than paying such unfair pre- miums. Similarly, increasing the deductible in a policy may sometimes be a financially attractive option.
These features of deductibles in insurance policies are illustrated by the choices your authors make. For example, both of their automobile policies offer either a $500 or a
$1,000 deductible associated with collision coverage. The
$500 deductible policy costs about $100 more each year.
Both authors have opted for the $1,000 policy on the princi- ple that paying $100 for an extra $500 coverage each year seems like a bad deal.
Homeowners’ policies offer a similar set of choices to your favorite authors. In this case, deductibles can be applied to both casualty and theft losses of property. Deductibles per occurrence of $500 are standard in these policies, and the discount for accepting a higher deductible ($1,000 or more) is very modest. Insurance companies do offer lower premiums for ‘‘claims-free’’ experience, however. This, in combination with the paperwork costs that filing a claim entails, may be sufficient to deter most claims under $1,000 anyway.
Deductibles in Health Insurance
Although the logic of a deductible applies to health insurance too, the presence of such features has proven to be quite controversial.1For example, in 1988 Congress passed the
Medicare Catastrophic Coverage Act. This act provided extra coverage for Medicare recipients, with a large annual deductible being required before coverage began. This policy proved unpopular for two reasons: (1) People argued that it was unfair to ask elderly people suffering ‘‘catastrophic’’ ill- nesses to pay the initial portion of their costs; and (2) the premium for the policy was to be paid by the elderly them- selves rather than by the working population (as is the case for a major portion of the rest of the Medicare program). The uproar over the program was so large that it was repealed after only one year.
More recently, arguments over deductibles surrounded the adoption of a Medicare drug benefit in 2003. Under the provisions of this plan (which came fully into effect in 2006), elderly consumers of prescription drugs would face a com- plex deductible scheme: (1) the first $250 spent annually on drugs is not covered by the drug benefit, (2) 75 percent of annual spending on drugs between $250 and $2,100 is covered by Medicare, (3) no spending between $2,100 and
$5,100 annually is covered by Medicare, and (4) 95 percent of annual spending over $5,100 is reimbursed by Medicare.
Observers have had a difficult time trying to find a rationale for such a complex scheme—especially for the odd ‘‘dough- nut hole’’ of coverage between $2,100 and $5,100 in annual spending. Clearly the provision cannot have much to do with the administrative cost issue. The $250 deductible at the bottom of the schedule prevents the filing of claims for every aspirin bought. It may be that the hole is intended mainly to save money so that available funds can be focused on the most needy elderly (those with drug expenses over
$5,100), but whether it has a rationale in the theory of insur- ance is anyone’s guess.
TOTHINKABOUT
1. In some cases, you can buy another insurance policy to cover a deductible in your underlying insurance. That is the case, for example, when you rent a car and for ‘‘Medi- gap’’ policies that cover Medicare deductibles. Does buying such a policy make sense?
2. Why are deductibles usually stated on an annual basis? If losses occur randomly, wouldn’t a ‘‘lifetime’’ deductible be better?
1Many health insurance policies also have ‘‘co-payment’’ provisions that require people to pay, say, 25 percent of their claim’s cost. Co- payments increase the price people pay for health careat the margin.
Deductibles reduce the average price paid, but, after the deductible is met, the marginal price of added care is zero. For a discussion of co-payments in health (and other) insurance, see Chapter 15.
The behavior of individuals after they are insured may also affect the possibility for insurance coverage. If having insurance makes people more likely to incur losses, insurers’ premium calculations will be incorrect and again, they may be forced to charge premiums that are too unfair in an actuarial sense. For example, after buying insurance for ski equipment, people may begin to ski more recklessly and treat the equipment more roughly because they no longer bear the cost of damage. To cover this increased chance of damage, insurance premiums may have to be very high.
Thismoral hazard in people’s behavior means that insurance against accidental losses of cash will not be available on any reasonable terms. In Chapter 15, we explore both adverse selection and moral hazard in much more detail.
Diversification
A second way for risk-averse individuals to reduce risk is by diversifying. This is the economic principle underlying the adage, ‘‘Don’t put all your eggs in one basket.’’
By suitably spreading risk around, it may be possible to raise expected utility above that provided by following a single course of action. This possibility is illustrated in Figure 4.3, which shows the utility of income for an individual with a current income of $35,000 who must invest $15,000 of that income in risky assets.
F I G U R E 4 . 3
D i ve r s i f i c at io n Re d u ce s R i s k
D U
C E U2
U1
Income (thousands of dollars)
0 20 35 50
Utility
Here, an investor must invest $15,000 in risky stocks. If he or she invests in only one stock, utility will beU1. Although two unrelated stocks may promise identical returns, investing in both of them can, on average, reduce risk and raise utility toU2.
For simplicity, assume there are only two such assets, shares of stock in company A or company B. One share of stock in either company costs $1, and the investor believes that the stock will rise to $2 if the company does well during the next year; if the company does poorly, however, the stock will be worthless.
Each company has a 50-50 chance of doing well. How should this individual invest his or her funds? At first, it would seem that it does not matter since the two companies’ prospects are identical. But, if we assume the company’s prospects are unrelated to one another, we can show that holding both stocks will reduce this person’s risks.
Suppose this person decides to plunge into the market by investing only in 15,000 shares of company A. Then he or she has a 50 percent chance of having
$50,000 at the end of the year and a 50 percent chance of having $20,000. This undiversified investment strategy will therefore yield a utility ofU1.
Let’s consider a diversified strategy in which the investor buys 7,500 shares of each stock. There are now four possible outcomes, depending on how each company does. These are illustrated in Table 4.1 together with the individual’s income in each of these eventualities. Each of these outcomes is equally likely.
Notice that the diversified strategy only achieves very good or very bad results when both companies do well or poorly, respectively. In half the cases, the gains in one company’s shares balance the losses in the other’s, and the individual ends up with the original $35,000. The diversified strategy, although it has the same expected value ($35,000¼0.25Æ$20,000þ0.50Æ$35,000þ0.25Æ$50,000) as the single-stock strategy, is less risky.
Illustrating the utility gain from this reduction in risk requires a bit of ingenuity because we must average the utilities from the four outcomes shown in Table 4.1.
We do so in a two-step process. PointCin Figure 4.3 represents the average utility for the case where company B does poorly (the average of the utility from $20,000 and $35,000), whereas pointDrepresents the average utility when company B does well ($35,000 and $50,000). The final average of pointsCandDis found at point E, which represents a utility level ofU2. BecauseU2exceedsU1, it is clear that this individual has gained from diversification.
The conclusion that spreading risk throughdiversificationcan increase utility applies to a number of situations. The reasoning in our simple illustration can be used, for example, to explain why individuals opt to buy mutual funds that invest in many stocks rather than choosing only a few stocks on their own (see Application 4.3: Mutual Funds). It also explains why people invest in many kinds of assets (stocks, bonds, cash, precious metals, real estate, and durable goods such as auto- mobiles) rather than in only one. The principle of diversification applies to spheres
T A B L E 4 . 1
P o ss i b l e O u t c o m e s f r o m I n v e s t i n g i n T w o C o mp a n i e s
COMPANY B’S PERFORMANCE
POOR GOOD
Company A’s Performance
Poor $20,000 $35,000
Good 35,000 50,000
Diversification The spreading of risk among several alternatives rather than choosing only one.